CLUSTERING MUTUAL FUNDS BY RETURN AND RISK LEVELS

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CLUSTERING MUTUAL FUNDS BY RETURN AND RISK LEVELS Francesco Lisi Edoardo Otranto

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CENTRO RICERCHE ECONOMICHE NORD SUD (CRENOS) UNIVERSITÀ DI CAGLIARI UNIVERSITÀ DI SASSARI

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Titolo: CLUSTERING MUTUAL FUNDS BY RETURN AND RISKL EVELS

Clustering mutual funds by return and risk levels∗ Francesco Lisi Dipartimento di Scienze Statistiche, Universit`a di Padova, Via Cesare Battisti 241/243, 35121 Padova, Italy e-mail: [email protected] Edoardo Otranto Dipartimento di Economia, Impresa e Regolamentazione and CRENoS, Universit`a di Sassari, Via Torre Tonda 34, 07100 Sassari, Italy e-mail: [email protected]

Abstract Mutual funds classifications, often made by rating agencies, are very common and sometimes criticized. In this work, a three-step statistical procedure for mutual funds classification is proposed. In the first step time series funds are characterized in terms of returns. In the second step, a clustering analysis is performed in order to obtain classes of homogeneous funds with respect to the risk levels. In particular, the risk is defined starting from an Asymmetric Threshold-GARCH model aimed todescribe minimum, normal and turmoil risk. The third step merges the previous two. An application to 75 European funds belonging to 5 different categories is given. Keywords: Cluster, distance, GARCH models, risk JEL Codes: C22, G11, G23

∗

We are grateful to the participants of the MAF 2008 conference, in particular to Alessandra Amendola, Giuseppe Storti, Umberto Triacca. This work was supported by Italian MIUR under Grant 2006137221 001.

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1 Introduction The number of mutual funds has grown dramatically over the last years. This has led to a number of classification schemes that should give reliable information to investors on features and performance of funds. Most of these classifications are produced by national or international rating agencies. For example, Morningstar groups funds into categories according to their actual investment style, portfolio composition, capitalization, growth prospects, etc. This information is then used, together with that related to returns, risks and costs, to set up a more concise classification commonly referred as Star Rating (see [11] for details). Actually, each rating agency has a specific owner evaluation method and also national associations of mutual funds managers keep and publish their own classifications. Problems arise as, in general, classes of different classifications do not coincide. Also, all classification procedures have some drawback; for example, they are often based on subjective information and require long elaboration time (see, for example, [15]). In the statistical literature, classification of financial time series received only relatively small attention. In addition, to the best of our knowledge, there are not comparisons among different proposed classifications and those of the rating agencies. Some authors use only returns for grouping financial time series. For example, [15] propose a classification scheme which combines different statistical methodologies (principal component analysis, clustering analysis, Sharpe’s constrained regression) applied on past returns of the time series. Also the clustering algorithm proposed by [9], referring to different kinds of functions, is based only on return levels. Other authors based their classifications only on risk and grouped the assets according to the distance between volatility models for financial time series ([12], [2], [13], [14], [8]). Risk-adjusted returns, i.e. returns standardized through standard deviation, are used for clustering time series by [4]. This approach is interesting, but using the unconditional variance as a measure of risk and ignoring the dynamics of volatility seems too simplistic. In this paper, a classification based only on the information contained in the net asset value (NAV) time series is considered. It rests on the simple and largely agreed idea that two very important points in evaluation of funds are return and risk levels. In order to measure the return level, the mean annual net period return is considered. As regards the riskiness, in the time series literature, it is commonly measured in terms of conditional variance (volatility) of a time series. As well known, volatility is characterized by a time-varying behavior and clustering effects, which imply that quiet (low volatility) and turmoil (high volatility) periods alternate. In order to account both for the time-varying nature of volatility and for its different behavior in quiet and turmoil periods, an asymmetric version of the standard Threshold GARCH model ([5], [17]), is considered in this work. The whole classification scheme consists of three steps: the first groups funds with respect to returns whereas the second with respect to riskiness. In particular, the whole risk is decomposed in constant minimum risk, time-varying standard risk and time-varying turmoil risk. Following [12], [13] and [14], the clustering related to volatility is based on a distance between GARCH models, which is an extension of the AR metric introduced by ([16]). Lastly, the third step merges the results of the first two steps to obtain a concise classification.

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The method is applied to 75 funds belonging to five categories: aggressive balanced funds, prudential balanced funds, corporate bond investments, large capitalization stock funds, monetary funds. In order to make a comparison with the classification implied by the Morningstar Star Rating, which ranges from 1 to 5 stars, our clustering is based on 5 “stars” as well. As expected, our classification does not coincide with the Morningstar Rating because it is only partially based on the same criteria. Nevertheless, in more than 82% of the considered funds the two ratings do not differ for more than one star. The paper is organized as follows. Section 2 describes how the risk is defined. Section 3 contains an application and the comparison of our clustering with the Morninstarg Rating classification. Section 4 concludes.

2

Risk modelling

In this section the reference framework for fund riskiness modelling is described. Let yt be the time series of the NAV of a fund and rt the corresponding log-return time series. We suppose that the return dynamics can be described by the following model: 1/2 rt = µt + εt = µt + ht ut , t = 1, ..., T (2.1) εt | It−1 ∼ N (0, ht ) where µt = Et−1 (rt ) is the conditonal expectation and ut is an i.i.d. zero-mean and unit variance innovation. The conditional variance ht follows an asymmetric version of the Threshold GARCH(1,1) process ([5], [17]), which stresses the possibility of a different volatility behavior in correspondence of high negative shocks. We refer to it as Asymmetric Threshold GARCH (AT-GARCH) model. Formally, the conditional variance can be described as: ht = γ + αε2t−1 + βht−1 + δSt−1 ε2t−1 ½ St =

1 0

if εt < ε∗t otherwise

(2.2)

where γ, α, β, δ are unknown parameters, whereas ε∗t is a threshold identifying the turmoil state. The value of ε∗t could represent a parameter to be estimated, but in this work we set it equal to the first decile of the empirical distribution of ε. On the whole, this choice maximizes the likelihood and the number of significant estimates of δ. Also, the first decile seems suitable because provides, through the parameter δ, the change in the volatility dynamics when high - but not extreme - negative returns occur. The purpose of this work is to classify funds in terms of gain and risk. While the net period return is the most common measure of gain, several possible risk measures are used in literature. However, most of them look at specific aspects of riskiness: standard deviation gives a medium constant measure; Value-at-Risk tries to estimate an extreme risk; the time-varying conditional variance in a standard GARCH model focuses on the time-varying risk, and so on. In this paper we make an effort to jointly looking at risk from different points of view. To do this, following [13], we consider the squared disturbances ε2t as a proxy

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of the instantaneous volatility of rt . It is well known that ε2t is a conditionally unbiased, but very noisy, estimator of the conditional variance and that realized volatility and intra-daily range are, in general, better estimators ([1], [3], [10]). However, the adoption of ε2t in our framework is justified by practical motivations because intradaily data are not available for mutual funds time series and, thus, realized volatility or range are not feasible. Starting from (2.2), after simple algebra, it can be shown that, for an AT-GARCH(1,1), ε2t follows the ARMA(1, 1) model: ¡ ¢ ¡ ¢ ε2t = γ + (α + δSt−j + β) ε2t−1 − β ε2t−1 − ht−1 + ε2t − ht , (2.3) where (ε2t − ht ) are uncorrelated, zero-mean errors. The AR(∞) representation of (2.3) is: ∞

ε2t

X ¡ ¢ γ = + (α + δSt−j )β j−1 ε2t−j + ε2t − ht , 1−β

(2.4)

j=1

from which it is easy to derive the expected value at time t given past information ∞

Et−1 (ε2t )

X γ + (α + δSt−1 )β j−1 ε2t−j . = 1−β

(2.5)

j=1

This representation splits the expected volatility, Et−1 (ε2t ), considered as a whole measure of risk, in three positive parts: a constant part, γ/(1 − β), representing the minimum which can be reached given the model; the standard P time-varyingj−1 P risk level j−1 ε2 ) and the time-varying turmoil risk ( ∞ (δS )β ε2t−j ), αβ risk ( ∞ t−j t−j j=1 j=1 the last two being dependent on past information. Of course, the estimation of expression (2.5) requires a finite truncation. In order to classify funds with respect to all these three risk components, we propose considering the distance between an homoskedastic model and a GARCH(1,1) model. Using the metric introduced by [12] and re-considered by [14], in the case of specification (2.2) this distance is given by: α + δSt−1 p . (1 − β 2 )

(2.6)

The previous analytical formulation allows us to provide a vectorial description of the risk of each fund. In particular, we characterize the minimum constant risk through the distance between the zero-risk case (γ = α = β = δ = 0) and the α = δ = 0 case γ vm = . (2.7) 1−β The time-varying standard risk is represented, instead, by the distance between a GARCH(1,1) model (δ = 0) and the corresponding homoskedastic model (α = β = δ = 0) α vs = p . (2.8) (1 − β 2 ) Lastly, the turmoil risk is described by the difference of the distance between an AT-GARCH model, and the homoskedastic model and the distance measured by (2.8): δ vt = p . (2.9) (1 − β 2 )

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The whole risk is then characterized by the vector [vm , vs , vt ]0 . If an extraelement, accounting for the return level, r¯, is considered, each fund may be featured by the vector: f = [¯ r, vm , vs , vt ]0 . In order to obtain groups of funds with similar return and risk levels, some clustering algorithm can be easily applied directly to f or to some function of the elements of f . For example, in the next section risk will be defined as the average of vm , vs and vt .

3

An application

As an application of the previous described procedure, the daily time series of NAV of 75 funds of the Euro area and belonging to five different categories were considered. The five typologies are the aggressive balanced, prudential balanced, corporate bond investments, large capitalization stock and monetary funds. Data, provided by Bloomberg, range from 1/1/2002 to 18/2/2008, for a total of 1601 observations for each series. Our experiment consists in providing a classification of these funds, characterizing each group in terms of return and riskiness (following the definitions of constant minimum, time-varying standard and time-varying turmoil risk) and comparing our classification with that produced by the Morningstar star rating. For each fund the return time series was considered and for each calendar year the net percentage return was computed; finally the average of the one-year returns, r¯, was used to represent the gain. To describe riskiness, first model (2.1) - (2.2) was estimated for each fund. When parameters were not significant at 5% level, they were set equal to zero and the corresponding constrained model was estimated. Of course, before accepting the model the absence of residual ARCH effects in the standardized residuals was checked. Parameter estimation allowed us to calculate the risks defined as in (2.7), (2.8) and (2.9) and to characterize the funds by the elements r¯, vm , vs , vt or by some their function. With these vectors a clustering analysis was performed. In the clustering, a classical hierarchical algorithm with the Euclidean distance was used, whereas distances between clusters are calculated following the average-linkage criterion (see, for example, [7]).1 In particular, the classification procedure followed three steps: 1. the series were classified in three groups, referring only to the degree of gain, i.e r¯ low, medium and high; 2. the series were classified in three groups only with respect to the degree of risk (low, medium and high). To summarize the different kinds of risk, the average of the three standardized risks was computed for each series. Standardization is important because of the different magnitudes of risks; for example, minimum risk generally has an order of magnitude lower than that of the two other risks; 3. the previous two classifications were merged combining the degree or gain and risk so as to obtain a rating from 1 to 5 “stars”; in particular, denoting with 1

The clustering was performed also using the Manhattan distance and the single-linkage and completelinkage criteria. Results are very similar.

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h, m and l the high, medium and low levels respectively and with the couple (a, b) the levels of gain and risk (with a, b = h, m, l), stars were assigned in the following way: 1 star for (l, h) (low gain and high risk); 2 stars for (l, m), (l, l) (low gain and medium risk, low gain and low risk); 3 stars for (m, h), (m, m), (h, h) (medium gain and high risk, medium gain and medium risk, high gain and high risk); 4 stars for (m, l), (h, m) (medium gain and low risk, high gain and medium risk); 5 stars for (h, l) (high gain and low risk). Of course, this is a subjective definition of stars, nevertheless it seemed reasonable to us. As in [13], the quality of the clustering was measured using the C-index ([6]). This index assumes values in the interval [0, 1], assuming small values when the quality of the clustering is good. In our experiments, we always obtained C ≤ 0.1. Table 1 lists the step-by-step results of the classification procedure for the group of monetary funds. The left part of the table shows the classification based on the risk evaluation and the global rating provided by Morningstar. The central part lists the elements characterizing the funds (one-year average return, constant minimum, timevarying standard and time-varying turmoil risks). Note that vm assumes very small values (due to the small values of γˆ ) and that only the last fund presents a turmoil risk.2 The right part of the table shows the results of the three-step classification procedure. The Gain column contains the classification in high, medium and low gain obtained by the clustering of step 1; the Risk column contains the classification in high, medium and low risk obtained by the grouping of step 2; lastly, the Stars column shows the five-group classification described in step 3. The differences with respect to the Morningstar rating are not large: the classification is the same in 8 cases over 15, in 6 cases it does not differ for more than one star and only in one case (the 14th fund) the two classifications differ for 2 stars. The same procedure was applied to the other four categories and results are summarized and compared with the Morningstar classification in Table 2. Clearly, the classifications are different because they are based on different criteria and definitions of gain and risk. However, in 82.7% of cases the two classifications do not differ for more than one star. This is evident looking at Table 3, in which the empirical probability function of the differences in stars and the corresponding cumulative distribution function are shown. Moreover, excluding the Corporate Bond Investments, which present the largest differences between the two classifications, the percentage of differences equal to or less than 1 increases up to 90% while the remaining 10% differs by two stars. In particular, the classifications relative to the Aggressive Balanced and the Monetary funds seem very similar between the two methodologies.

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Some concluding remarks

In this paper a clustering procedure to classify mutual funds in terms of gain and risk has been proposed. It refers to a purely statistical approach, based on few tools 2

On the whole, instead, parameter δ resulted significant in 11 cases (about the 14% of funds).

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Table 1: Monetary funds: Morningstar classification and details of the clustering procedure. Morningstar Clustering Risk

Stars Return

low below average below average below average below average below average average average average above average above average above average above average high high

3 5 3 3 3 3 2 2 4 2 4 2 3 5 1

2.25 2.66 2.08 2.26 2.34 2.26 1.70 1.87 2.41 2.05 2.71 2.10 1.96 2.28 1.80

vm

vs

4.77E-09 0 7.01E-08 5.71E-08 0 0 1.93E-07 0 0 0 1.91E-07 0 0 1.29E-06 0

vt

0 0 0.087 0 0.171 0 0 0 0.180 0 0.231 0 0 0 0.144 0 0.208 0 0.155 0 0.385 0 0.234 0 0.145 0 0 0 0.151 0.333

Gain

Risk

Stars

medium high low medium medium medium low low medium low high low low medium low

low low medium low medium medium low medium medium medium high medium medium medium high

4 5 2 4 3 3 2 2 3 2 3 2 2 3 1

Table 2: Comparison of Morningstar and Clustering Classification. differences in stars Stars Aggr. Bal.

Clustering Morningstar Prud. Bal. Clustering Morningstar Corp. Bond Clustering Morningstar Stock Clustering Morningstar Monetary Clustering Morningstar

1 2 3

4 5 0 1 2 3

2 0 0 0 1 0 0 1 1 1

1 4 7 4 2 2 11 5 2 2

7

4 3 2 2 0 3 1 3 6 4

7 8 6 9 3 10 2 6 5 6

1 0 0 0 9 0 1 0 1 2

7 6 2 4 9 2 3 5 5 2 4 10 1 8 6 1

Table 3: Empirical probability and cumulative distribution functions of differences in stars (percentages). Empirical probability function 0 1 2 3 4 5 34.7 48.0 14.7 2.6 0.0 0.0 Empirical cumulative distribution function 0 1 2 3 4 5 34.7 82.7 97.4 100 100 100

to characterize return and risk. The method is model-based, in the sense that the definition of risk is linked to the estimation of a particular Threshold GARCH model, which characterizes quiet and turmoil states of financial markets. The risk is evaluated simply considering an equally-weighted average of three different kinds of risk (constant minimum risk, time-varying standard risk and timevarying turmoil risk). Different weights could also be considered but at the cost of introducing a subjectivity element. Surprisingly, in our application, this simple method provided a classification which does not show large differences with respect to the Morningstar classification. Of course, this exercise could be extended to compare our clustering method with other alternative classifications. A very interesting application would be using this approach in asset allocation or portfolio selection problems.

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[8] Kalantzis, T., Papanastassiou, D.: Classification of GARCH time series: an empirical investigation. Applied Financial Economics 18 1–6 (2008) [9] Lytkin, N.I., Kulikowski, C.A., Muchnik, I.B.: Variance-based criteria for clustering and their application to the analysis of management styles of mutual funds based on time series of daily returns. DIMACS Technical Report 200801 (2008) [10] Martens M., van Dijk D.: Measuring volatility with realized range. Journal of Econometrics 138, 181–207 (2007) [11] Morningstar: The Morningstar rating methodology. Morningstar Methodology Paper (2007). [12] Otranto, E.: Classifying the markets volatility with ARMA distance measures. Quaderni di Statistica 6 (2004) 1–19. [13] Otranto, E.: Clustering heteroskedastic time series by model-based procedures. Computational Statistics and Data Analysis (2008) doi:10.1016/j.csda.2008.03.020 [14] Otranto, E., Trudda, A.: Evaluating the Risk of Pension Funds by Statistical Procedures. In: Lakatos, G.M. (ed.): Transition Economies: 21st Century Issues and Challenges, pp. 189-204. Nova Science Publishers, Hauppauge (2008) [15] Pattarin, F., Paterlini, S. and Minerva, T.: Clustering financial time series: an application to mutual funds style analysis. Computational Statistics and Data Analysis, 47, 353–372 (2004) [16] Piccolo, D.: A distance measure for classifying ARIMA models. Journal of Time Series Analysis 11, 153–164 (1990) [17] Zakoian, J.M.: Threshold heteroskedastic models. Journal of Economic Dynamic and Control 18, 931-955 (1994)

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